# Quantum Tamo-Barg (QTB) code[1]

## Description

A member of a family of Galois-qudit CSS codes whose underlying classical codes consist of Tamo-Barg codes together with specific low-weight codewords. Folded versions of QTB codes, or FQTB codes, defined on qudits whose dimension depends on \(n\) yield explicit examples of QLRCs of arbitrary locality \(r\) [1; Thm. 2].

## Protection

A family of QTBs can be defined for every prime \(r\), rate \(R\in(0,1)\), and qudit dimension \(q = n+1\) such that their relative distance is \(\delta \geq 1 - \sqrt{(1+R)/2} - O(1/r)\) [1; Thm. 3].

Folding these codes by combining qudits into larger qudits yields FQTB codes with relative distance \(\delta \geq (1-R)/2 - O(1/\sqrt{r})\) [1; Thm. 2] and qudit dimension to \(q = n^{O(r^2)}\). This relative distance is of order \(O(1/\sqrt{r})\) below the Singleton-like QLRC bound.

## Decoding

Polynomially efficient decoder for QTB codes against errors acting on a number of subsystems that can go up to half of their conjectured distance [1; Thm. 8]. The decoder is based on decoding RS codes, and its runtime is independent of the locality \(r\).Polynomially efficient decoder for FQTB codes against errors acting on a number of subsystems that can go up to half of their conjectured distance [1; Thm. 7]. The runtime depends on the locality \(r\).

## Parents

- Galois-qudit CSS code
- Quantum locally recoverable code (QLRC) — Folded versions of QTB codes defined on qudits of dimension \(q = n^{O(r^2)}\) yield explicit examples of QLRCs of arbitrary locality \(r\) [1; Thm. 2].

## Cousin

- Tamo-Barg code — QTB codes are CSS codes constructed from Tamo-Barg codes.

## References

- [1]
- L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653

## Page edit log

- Victor V. Albert (2024-03-26) — most recent

## Cite as:

“Quantum Tamo-Barg (QTB) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_tamo_barg